Exam 5: Final Examination

You may use a calculator for numerical calculations, but not for symbolic computation.

You may use your note cards from Exams 1-3 and one new 3"-by-5" note card. The p. 161 table of generating functions and the filled out Table of Discrete Distributions will be provided.

The final exam will focus on recent material, but some problems or subproblems will relate to earlier topics.

RECENT TOPICS

• Generating functions
  • "THE MOST POWERFUL WAY to deal with sequences of numbers, as far as anybody knows, is to manipulate infinite series that 'generate' those sequences." --Graham, Knuth & Patashnik
  • Ordinary generating functions (ogf)
    Two perspectives:
    • Formal power series
    • Calculus-defined power series
  • Basic manipulations
    They're like polynomials of unbounded degree.
  • Dictionary of common ogfs: see esp. p. 161.
    See also formulas 1, 2, 6, 7 of the top ten summation formulas
  • Solving counting problems via generating functions
    Computations can be turned over to Maple.
  • Solving recurrence relations via generating functions
• Recurrence relations
  • First-order nonhomogeneous linear recurrence relations (with possibly variable coefficients) (This is an older topic, actually, but relevant.)
    • Solution by summation factor
    • Solution by unwinding/unfolding/backtracking
  • Linear recurrence relations with constant coefficients
    • Solution via characteristic polynomials
      • Homogeneous case
      • Special nonhomogeneous cases
    • Solution via generating functions
  • Divide-and-conquer algorithms and recurrence relations
    • f(n) = a f(n/b) + g(n)
    • Master Theorem

  Here's an old MCS 256 exam and my solutions. Here's an old MCS 256 exam on generating functions and recurrence relations. (Sorry, I don't have the answers stored electronically.)

  PREVIOUS TOPICS

  See the summaries and documents provided on the course web site and throughout the semester
  • Recurrences and mathematical induction
  • Finite difference calculus
  • Summation calculus
  • Asymptotics
  • Combinatorics
    • How to count
    • Intermediate counting techniques
  • Discrete probability
    • Probability spaces, conditional probability
    • Random variables and distributions
      Special discrete distributions
    • Moments: mean, variance, etc.; probability generating function
    • Joint distributions
  • Generating functions
  • Recurrence relations